Drift & Meander of Spiral Waves

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Spiral Waves Theory of Drift and Meander of Spiral Waves Spiral Wave Solutions in the Comoving Frame of Reference Concluding Remarks

Andy Foulkes & Vadim Biktashev

12 March 2009

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Theory of Drift and Meander of Spiral Waves Spiral Wave Solutions in the Comoving Frame of Reference Concluding Remarks Table of contents 1 Spiral Waves Introduction Spiral Wave Motion Models Studied General Comments 2 Theory of Drift and Meander of Spiral Waves General Approach Example: Electrophoresis Induced Drift Drift & Meander 3 Spiral Wave Solutions in the Comoving Frame of Reference General Idea & Motivation Rigid Rotation- Stationary Solutions Meandering Waves - Periodic Solutions Applications - 1:1 Resonance & Large Core Spirals 4 Concluding Remarks

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves - Introduction

Spiral waves occur naturally in physical (galaxies, liquid crystals), chemical (rusting, BZ reaction) and biological systems (cardiac arrhythmias, Dictyostelium Discoideum) One of the most famous examples of spiral waves is in the Belousov-Zhabotinski (BZ) reactions (circa 1960) Popularised in the West by Arthur Winfree in the 1970’s First ﬁnite automata model was derived in 1946 by Wiener & Rosenblueth Since then, they have been the subject of intense research involving many interdisiplinary teams.

Spiral Waves - BZ Reaction Zhabotinsky A. M. and Zaikin, 1971

Andy FoulkesFigure: & Vadim Snapshot Biktashev ofDrift a & spiral Meander wave of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Project Overview

Dynamical Systems project - not much Biology My project commenced as a project to study the uniﬁcation of the two main types of motion of spiral waves - Drift and Meander Frequency locking Numerical technique to study spiral waves in a comoving frame of reference (came from studies into frequency locking) Used response functions

Three main types of motion of the spiral wave that we consider:

Rigid Rotation - shape of the arm of the spiral is ﬁxed

Meander - quasiperiodic motion, arm changes shape

Drift - motion about a moving center of rotation

For the theory, we ﬁrst consider just a general Reaction-Diﬀusion system of equations:

∂u 2 n = D u + f(u) + h(u, u, r, t) u = (u1, u2,..., un) R ∂t ∇ ∇ ∈

Numerical work involves Barkleys and FitzHugh-Nagumo (FHN) models: Barkley’s FHN

3 1 u2+b 1 u1 fu1 (u1, u2) = ε u1(1 − u1)(u1 − a ) fu1 (u1, u2) = ε (u1 − 3 − u2) fu2 (u1, u2) = (u1 − u2) fu2 (u1, u2) = ε(u1 + β − γu2)

Reaction-Diﬀusion equations generating spiral wave solutions are equivariant under Euclidean Symmetry (group SE(2) of rotations and translations)

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Approach

There are two separate theories for Meander of Spiral Waves and Drift of Spiral Waves: 1995 - Biktashev & Holden - “Resonant Drift of Autowave Vortices and the Eﬀects of Boundaries and Inhomogeneities” 1996 - Biktashev, Holden & Nikolaev - “Spiral Wave Meander and Symmetry of the Plane” The aim of this work is to unify the two theories and determine the dynamics of spiral waves which both drift and meander simultaneously by studying the equations of motion of the tip of the spiral wave Problem - Theories written using diﬀerent techniques Solution - Rewrite one theory using techniques similar to the the other

Once theory of drift had been rewritten, study meander Note that in the space of group orbits we have limit cycle solutions Use Floquet Theory to study these limit cycles Find explicit analytical solutions in the space of group orbits

We will exploit the fact that the system generating Spiral Wave solution is equivariant under Euclidean Symmetry We therefore use Group Theory techniques and consider group elements g SE(2), where g = (R, Θ) such that R is a translation in∈ the (x, y)-plane with R = (X , Y ) and Θ is a rotation. We will also choose R and Θ such that they are the tip coordinates and orientation.

We ﬁrst consider a spiral wave solution to a Reaction-Diﬀusion system with a symmetry breaking perturbation:

∂u 2 n = D u + f(u) + h(u, u, r, t), u = u(r, t) R (1) ∂t ∇ ∇ ∈

Represent this in a Banach space - space of all group orbits

U ′ B G U

g g′

V M V ′

Figure: Representation in a Banach Space

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves

1 Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves (cont.)

Split out motion along group orbits and Representative Manifold We ﬁnd that the associated Reaction-Diﬀusion-Advection system is:

∂v 2 ∂v n = D v + f(v) + (c, )v + ω + h˜(v, u, r, t, g), v R (2) ∂t ∇ ∇ ∂θ ∇ ∈ v1(0, 0) = u∗ (3)

v2(0, 0) = v∗ (4) ∂v 1 (0, 0) = 0 (5) ∂x

h˜(v, u, r, t, g) = T(g −1)h(T(g)v, T(g) u, r, t) ∇ ∇ Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves (cont.)

The solutions to this system of equations are ( = (v, c, ω)), where v is the spiral wave solution in a frameM of reference comoving with the tip of the spiral wave. c is the translation velocity of the comoving frame of reference, and ω is its rotational velocity. Frame of reference ﬁxed to the tip of the spiral wave via the tip pinning conditions (Eqns.(3)-(5)).

The tip equations of motion are:

dR = ceiΘ (6) dt dΘ = ω (7) dt Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves (cont.)

We now apply perturbation techniques as follows: 2 v = v0 + v1 + O( ) (8) 2 cx = c0x + c1x + O( ) (9) 2 cy = c0y + c1y + O( ) (10) 2 ω = ω0 + ω1 + O( ) (11)

which yields the following:

∂v0 2 = 0 = D v0 + f(v0) + (c0, )v0 + ω0∂θv0 ⇒ ∂t ∇ ∇ ∂v1 2 df(v0) = 0 = D v1 + v1 + (c0, )v1 + ω0∂θv1 6 ⇒ ∂t ∇ dv · ∇ +(c1, )v0 + ω1v0 + h˜(v0, r, t) + O() ∇

Now let: 2 df(v0) Lv1 = D v1 + v1 + (c0, )v1 + ω0∂θv1 (12) ∇ dv · ∇ and hˆ = (c1, )v0 + ω1∂θv0 + h˜(v0, r, t) (13) ∇

which gives us a linear approximation for the perturbed part of the spiral wave:

∂v 1 = Lv + hˆ + O() (14) ∂t 1

Now, L is a linear operator and so satisﬁes the following eigenvalue problem:

Lφi = λi φi (15)

Also, we have that the adjoint eigenvalue problem is:

+ ¯ L ψj = λj ψj (16)

ψj are the Response Functions.

For rigidly rotating spiral waves, there are three critical eigenvalues (i.e. eigenvalues which have zero real part):

λ±1 = iω ± λ0 = 0

These eigenvalues are directly related to the symmetry of the problem ω is the angular velocity of the spiral wave

ˆ (ψ0,±1, h) = 0 (17)

and also the orthogonality conditions:

(ψj , φi ) = δij (18)

The scalar products (α, β) are deﬁned as:

IZ ∞ (α, β) = α, β drdθ (19) 0 h i

We ﬁnd that the equations of motion of the tip of the spiral wave are now:

dR iΘ iΘ ¯ ˆ c0 ˆ = c0e e 2(ψ1, h) + (ψ0, h) (20) dt − ω0 dΘ = ω0 + (ψ , hˆ) (21) dt 0

where ψ0 and ψ1 are response functions.

We take a perturbation of the form:

∂v h = A 0 (22) ∂x

where A is an n n matrix of the form: ×   A1 0 0 ··  0 A2 0   ··  A = [Aij ] =  0 A3   · ··   0  ···· 0 0 An ·· and each Aij is small and is represented as Aij = aij .

We now need to consider the transformed perturbation h˜:

∂v ∂v h˜ = A cos(Θ) 0 (r) sin(Θ) 0 (r) ∂x − ∂y

Going back now to the equations of motion, we now have:

dR iΘ iΘ ∂v0 ∂v0 = c0e e 2 ψ¯ , A cos(Θ) (r) sin(Θ) (r) dt − 1 ∂x − ∂y c0 ∂v0 ∂v0 + ψ0, A cos(Θ) (r) sin(Θ) (r) (23) ω0 ∂x − ∂y dΘ ∂v0 ∂v0 = ω0 + ψ , A cos(Θ) (r) sin(Θ) (r) (24) dt 0 ∂x − ∂y

which can be rewritten as:

„ « dR iΘ c0 2iΘ = c0e − A0,−1 + A−1,−1 e dt 2ω0 −A−1,1 (25)

dΘ 1 −iΘ iΘ = ω0 − (A0,1e + A0,−1e ) (26) dt 2

Solving these ODE’s we get:

ic ia 0 i(ω0t+Θ∗) −1,−1 2i(ω0t+Θ∗) R = R(0) e + e a−1,1t(27) − ω0 2ω0 − i −i(ω0t+Θ∗) i(ω0t+Θ∗) Θ = (ω0t + Θ∗) a0,1e + a0,−1e (28) − 2ω0

Y 20

0 5 10 15 20 25 X Figure: Comparison of the numerical simulation (left) to the analytical prediction (right)

Due to meander, c and ω are not constant In functional space, c and ω form limit cycles Use Floquet Theory

Found that a regular perturbation method may not work for some types of perturbations So, singular perturbation method needs to be used In the singular perturbation method, we deﬁne a new time variable as: τ = t + θ(t)

The equations of motion are now:

dR iΘ iΘ = c0(t)e e 2(ψ¯ (t + θ(t)), h˜(v, t)) dt − 1 c0(t) ˜ + (ψ0(t + θ(t)), h(v, t)) ω0(t) dΘ = ω0(t) + (ψ (t + θ(t)), h˜) dt 0 dθ = (ψ (t + θ(t)), h˜(v, t)) dt ∗

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Idea & Motivation Theory of Drift and Meander of Spiral Waves Rigid Rotation- Stationary Solutions Spiral Wave Solutions in the Comoving Frame of Reference Meandering Waves - Periodic Solutions Concluding Remarks Applications - 1:1 Resonance & Large Core Spirals Numerical Simulations - Solution in the Comoving frame

Solution of Spiral Waves in a comoving frame of reference can be advantageous One advantage is faster simulations - smaller box sizes Solve Reaction-Diﬀusion-Advection system - Advection terms move the frame of reference subject to tip pinning conditions

Consider system with no symmetry breaking perturbations (i.e. no externally forced drift considered).

∂v 2 ∂v n = D v + f(v) + (c, )v + ω , v R ∂t ∇ ∇ ∂θ ∈

Tip Pinning Tip Pinning Conditions 1 Conditions 2

v1(0, 0) = u∗ v1(0, 0) = u∗ v2(0, 0) = v∗ v2(0, 0) = v∗ ∂v1 ∂x (0, 0) = 0 v1(xinc , yinc ) = u∗

Rigid Rotation - time vs cx Rigid Rotation - time vs cy 0.428 0.019

0.426 0.018

0.424 0.017

0.422 0.016

0.42 0.015

cx 0.418 cy 0.014

0.416 0.013

0.414 0.012

0.412 0.011

0.41 0.01

0.408 0.009 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 time time

Rigid Rotation - time vs omega -0.606

-0.608

-0.61

-0.612

-0.614

-0.616 omega

-0.618

-0.62

-0.622

-0.624

-0.626 0 20 40 60 80 100 120 140 160 time Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Idea & Motivation Theory of Drift and Meander of Spiral Waves Rigid Rotation- Stationary Solutions Spiral Wave Solutions in the Comoving Frame of Reference Meandering Waves - Periodic Solutions Concluding Remarks Applications - 1:1 Resonance & Large Core Spirals Rigidly Rotating Wave

Given the numerical data for cx ,cy and ω, we can reconstruct the tip trajectory by numerically integrating the equations of motion:

dR = ceiΘ dt dΘ = ω dt

Rigid Rotation - trajectory Rigid Rotation Tip trajectory (FHN) - laboratory frame 15.2 16.2

15 16

14.8 15.8

14.6 15.6

Y 14.4 Y 15.4

14.2 15.2

14 15

13.8 14.8

13.6 14.6 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 13.6 13.8 14 14.2 14.4 14.6 14.8 15 15.2 X X

Figure: (left)Tip trajectory (from data in the comoving frame of reference, (right) Tip trajectory (Laboratory frame).

Meander (FHN) - cx vs cy Meander (FHN) - cx vs omega -0.1 -0.35

-0.4 -0.2

-0.45

-0.3 -0.5 cy omega -0.55 -0.4

-0.6

-0.5 -0.65

-0.6 -0.7 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 cx cx

Meander (FHN) - cy vs omega -0.3

-0.35

-0.4

-0.45

-0.5

omega -0.55

-0.6

-0.65

-0.7

-0.75 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 cy Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Idea & Motivation Theory of Drift and Meander of Spiral Waves Rigid Rotation- Stationary Solutions Spiral Wave Solutions in the Comoving Frame of Reference Meandering Waves - Periodic Solutions Concluding Remarks Applications - 1:1 Resonance & Large Core Spirals Meandering Wave

Meander (FHN) - trajectory Tip trajectory (FHN) - laboratory frame 19 19

18 18

17 17

Y 16 Y 16

15 15

14 14

13 13 13 14 15 16 17 18 19 9 10 11 12 13 14 15 X X

Figure: (left) Reconstructed trajectory from the quotient data, (right) tip trajectory from the laboratory frame of reference

Meandering Waves - 1:1 Resonance

two meandering frequencies are equal get what appears to be drift - but still meander with inﬁnite core radius problem - need very large box size to get meaningful results solution - study solution in a frame of reference comoving with the spiral Use Barkley Model

Figure: Barkley Parametric Portrait for = 0.02 (Barkley et al, PRL, 72(1), 1994)

a=0.5991, b=0.05, ε=0.02 a=0.6191, b=0.05, ε=0.02 40

35 10

0 25

20 -10 Y Y 15

10 -20

0 -30

-5 0 5 10 15 20 25 30 35 40 45 -20 -10 0 10 20 30 X X

Trajectory at 1:1 resonance 220

200

180

160

140

120 Y 100

0 -600 -500 -400 -300 -200 -100 0 100 X Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Idea & Motivation Theory of Drift and Meander of Spiral Waves Rigid Rotation- Stationary Solutions Spiral Wave Solutions in the Comoving Frame of Reference Meandering Waves - Periodic Solutions Concluding Remarks Applications - 1:1 Resonance & Large Core Spirals Meandering Wave - 1:1 Resonance

cx vs cy y c

cx Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Idea & Motivation Theory of Drift and Meander of Spiral Waves Rigid Rotation- Stationary Solutions Spiral Wave Solutions in the Comoving Frame of Reference Meandering Waves - Periodic Solutions Concluding Remarks Applications - 1:1 Resonance & Large Core Spirals Meandering Wave - 1:1 Resonance

ω cx vs ω

ω cy vs ω

cy Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Theory of Drift and Meander of Spiral Waves Spiral Wave Solutions in the Comoving Frame of Reference Concluding Remarks Conclusions

Conclusions: Used Group Theory and Perturbation Methods to study dynamics of Drifting Spiral Waves. Extended this to meander using Floquet theory. Developed a program EZ-Freeze which solves spiral waves in a comoving frame of reference. Study spiral waves in 1:1 resonance in comoving frame of reference, reducing computational time signiﬁcantly. For large core spirals, we conﬁrmed which asymptotic theory is correct.

Further Work: Apply numerical calculations of response functions to predict the drift of spiral waves using the theory developed in this project for rigidly rotating waves (for conﬁrmation) and also for meandering waves (new work). Develop further methods for detecting frequency locking and extend the theory developed so far to frequency locking (this has been initiated). Investigate the drift of meandering spiral waves using EZ-Freeze. Look at scroll wave meander. Also, look at applying the developed theory to simulations using more realistic models for, say, cardiac simulations.

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Theory of Drift and Meander of Spiral Waves Spiral Wave Solutions in the Comoving Frame of Reference Concluding Remarks

THANK YOU!!!

Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves